Multiple recurrence and convergence for hardy sequences of polynomial growth

We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for “typical” choices of Hardy field functions a(t) with polynomial growth, the averages

${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[a(n)]}}x) \cdots {f_\ell }({T^{\ell [a(n)]}}x)} $

converge in mean and we determine their limit. For example, this is the case if a(t) = t ^3/2, t log t, or t ² + (log t)². Furthermore, if {a ₁(t), …, a _? (t)} is a “typical” family of logarithmico-exponential functions of polynomial growth, then for every ergodic system, the averages

${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[{a_1}(n)]}}x) \cdots {f_\ell }({T^{[{a_\ell }(n)]}}x)} $

converge in mean to the product of the integrals of the corresponding functions. For example, this is the case if the functions a _i (t) are given by different positive fractional powers of t. We deduce several results in combinatorics. We show that if a(t) is a non-polynomial Hardy field function with polynomial growth, then every set of integers with positive upper density contains arithmetic progressions of the form {m,m + [a(n)], …, m + ?[a(n)]}. Under suitable assumptions, we get a related result concerning patterns of the form {m,m + [a ₁(n)], …,m + [a _? (n)]}.

     

Averaging of systems of differential inclusions with slow and fast variables

We consider the system of differential inclusions

$$\dot x \in \mu F(t, x, y, \mu ), x(0) = x_0 , \dot y \in G(t, x, y, \mu ), y(0) = y_0 $$

, where F,G: D → Kυ (\(R^{m_1 } \)), Kυ (\(R^{m_2 } \)) are mappings into the sets of nonempty convex compact sets in the Euclidean spaces \(R^{m_1 } \) and \(R^{m_2 } \), respectively, D = R ₊ × \(R^{m_1 } \) × \(R^{m_2 } \) × [0, a], a > 0, and µ is a small parameter. The functions F and G and the right-hand side of the averaged problem \(\dot u\) ∈ µF ₀(u), u(0) = x ₀, F ₀(u) ∈ Kυ (\(R^{m_1 } \)), satisfy the one-sided Lipschitz condition with respect to the corresponding phase variables. Under these and some other conditions, we prove that, for each ? > 0, there exists a µ > 0 such that, for an arbitrary µ ∈ (0, µ₀] and any solution x _µ(·), y _µ(·) of the original problem, there exists a solution u _µ(·) of the averaged problem such that ∥x _µ(t) ? y _µ(t) ∥ ≤ ? for t ∈ [0, 1/µ]. Furthermore, for each solution u _µ(·)of the averaged problem, there exists a solution x _µ(·), y _µ(·) of the original problem with the same estimate.

     

A positive solution of a nonlinear Schrdinger system with nonconstant potentials

In this paper, we study the existence of positive solutions to the following Schr¨odinger system:{-?u + V_1(x)u = μ_1(x)u~3+ β(x)v~2u, x ∈R~N,-?v + V_2(x)v = μ_2(x)v~3+ β(x)u~2v, x ∈R~N,u, v ∈H~1(R~N),where N = 1, 2, 3; V_1(x) and V_2(x) are positive and continuous, but may not be well-shaped; and μ_1(x), μ_2(x)and β(x) are continuous, but may not be positive or anti-well-shaped. We prove that the system has a positive solution when the coefficients Vi(x), μ_i(x)(i = 1, 2) and β(x) satisfy some additional conditions.     

Boundary value problems for complete quasi-parabolic differential equations of odd order with variable domains of operators

We prove the well-posed solvability in the strong sense of the boundary value Problems

$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$

where the unbounded operators A _s (t), s > 0, in a Hilbert space H have domains D(A _s (t)) depending on t, are subordinate to the powers A ^1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A ₀(t) and the symmetric operators A _s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that

$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$

, where the smoothing operators are defined by

$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$

.

     

On Templeman averages and variation functions

Let S be a countable semigroup acting in a measure-preserving fashion (g ? T _g ) on a measure space (Ω, A, µ). For a finite subset A of S, let |A| denote its cardinality. Let (A _k ) _k=1 ^∞ be a sequence of subsets of S satisfying conditions related to those in the ergodic theorem for semi-group actions of A. A. Tempelman. For A-measureable functions f on the measure space (Ω, A, μ) we form for k ≥ 1 the Templeman averages \(\pi _k (f)(x) = \left| {A_k } \right|^{ - 1} \sum\nolimits_{g \in A_k } {T_g f(x)}\) and set V _q f(x) = (Σ _k≥1|π _k+1(f)(x) ? π _k (f)(x)|q)^1/q when q ∈ (1, 2]. We show that there exists C > 0 such that for all f in L ¹(Ω, A, µ) we have µ({x ∈ Ω: V _q f(x) > λ}) ≤ C(∫_Ω | f | dµ/λ). Finally, some concrete examples are constructed.     

On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

An extended class of stabilizable uncertain systems

The system of equations \(\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u\), where A(·) ∈ ?^{n × n}, B(·) ∈ ?^{n × m}, S(·) ∈ R^{n × m}, is considered. The elements of the matrices A(·), B(·), S(·) are uniformly bounded and are functionals of an arbitrary nature. It is assumed that there exist k elements \({\alpha _{{i_i}{j_l}}}\left( \cdot \right)\left( {l \in \overline {1,k} } \right)\) of fixed sign above the main diagonal of the matrix A(·), and each of them is the only significant element in its row and column. The other elements above the main diagonal are sufficiently small. It is assumed that m = n ?k, and the elements β_ij(·) of the matrix B(·) possess the property \(\left| {{\beta _{{i_s}s}}\left( \cdot \right)} \right| = {\beta _0} > 0\;at\;{i_s}\; \in \;\overline {1,n} \backslash \left\{ {{i_1}, \ldots ,{i_k}} \right\}\). The other elements of the matrix B(·) are zero. The positive definite matrix H = {h_ij} of the following form is constructed. The main diagonal is occupied by the positive numbers h_ii = h_i, \({h_{{i_l}}}_{{j_l}}\, = \,{h_{{j_l}{i_l}}}\, = \, - 0.5\sqrt {{h_{{i_l}}}_{{j_l}}} \,\operatorname{sgn} \,{\alpha _{{i_l}}}_{{j_l}}\left( \cdot \right)\). The other elements of the matrix H are zero. The analysis of the derivative of the Lyapunov function V(x) = x*H^–1x yields h_i\(\left( {i \in \overline {1,n} } \right)\) and λ_i ≤ 0 \(\left( {i \in \overline {1,n} } \right)\) such that for S(·) = H^?1ΛB(·), Λ = diag(λ₁, ..., λ_n), the system of the considered equations becomes globally exponentially stable. The control is robust with respect to the elements of the matrix A(·).     

Meromorphic Jost functions and asymptotic expansions for Jacobi parameters

We show that the parameters a _n , b _n of a Jacobi matrix have a complete asymptotic expansion

$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $

, where 1 < |µ_j| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ^?1) is an entire meromorphic function. We relate the poles of u to the µ_j’s.

     

Stability of nonlocal difference schemes in a subspace

We consider a family of two-layer difference schemes for the heat equation with nonlocal boundary conditions containing the parameter γ. In some interval γ ∈ (1, γ ₊), the spectrum of the main difference operator contains a unique eigenvalue λ ₀ in the left complex half-plane, while the remaining eigenvalues λ ₁, λ ₂, …, λ _N?1 lie in the right half-plane. The corresponding grid space H _N is represented as the direct sum H _N = H ₀⊕H _N?1 of a one-dimensional subspace and the subspace H _N?1 that is the linear span of eigenvectors µ⁽¹⁾, µ⁽²⁾, …, µ^(N?1). We introduce the notion of stability in the subspace H _N?1 and derive a stability criterion.     

On exponential sums over primes and application in Waring-Goldbach problem

In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2 log κ/log2, x≥2 and α=a/q λsubject to (a, q) = 1, 1≤a≤q, and λ∈R. Then As an application, we prove that with at most O(N2/8 ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.     

Irreducible Modules Over Witt Algebras $\mathcal {W}_{n}$ and Over 𝖘𝖑n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$

In this paper, by using the “twisting technique” we obtain a class of new modules A _b over the Witt algebras \(\mathcal {W}_{n}\) from modules A over the Weyl algebras \(\mathcal {K}_{n}\) (of Laurent polynomials) for any \(b\in \mathbb {C}\). We give necessary and sufficient conditions for A _b to be irreducible, and determine necessary and sufficient conditions for two such irreducible \(\mathcal {W}_{n}\)-modules to be isomorphic. Since \(\mathfrak {sl}_{n+1}(\mathbb {C})\) is a subalgebra of \(\mathcal {W}_{n}\), all the above irreducible \(\mathcal {W}_{n}\)-modules A _b can be considered as \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules. For a class of such \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules, denoted by Ω_1?a (λ ₁, λ ₂, ? ,λ _n ) where \(a\in \mathbb {C}, \lambda _{1},\lambda _{2},\cdots ,\lambda _{n} \in \mathbb {C}^{*}\), we determine necessary and sufficient conditions for these \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules to be irreducible. If the \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module Ω_1?a (λ ₁, λ ₂,? ,λ _n ) is reducible, we prove that it has a unique nontrivial submodule W _1?a (λ ₁, λ ₂,...λ _n ) and the quotient module is the finite dimensional \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module with highest weight mΛ _n for some non-negative integer \(m\in \mathbb {Z}_{+}\). We also determine necessary and sufficient conditions for two \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules of the form Ω_1?a (λ ₁, λ ₂,? ,λ _n ) or of the form W _1?a (λ ₁, λ ₂,...λ _n ) to be isomorphic.     

Blue sky catastrophe in relaxation systems with one fast and two slow variables

We establish conditions under which three-dimensional relaxational systems of the form

$$\dot x = f(x,y,\mu ),\varepsilon \dot y = g(x,y),x = (x_1 ,x_2 ) \in \mathbb{R}^2 ,y \in \mathbb{R},$$

where 0 ≤ ε ? 1, |µ| ? 1, and f, g ∈ C ^∞, exhibit the so-called blue sky catastrophe [the appearance of a stable relaxational cycle whose period and length tend to infinity as µ tends to some critical value µ_*(ε), µ_*(0) = 0].

     

On almost good triples of vertices in edge regular graphs

Consider a connected edge regular graph Γ with parameters (v, k, λ) and put b ₁ = k?λ?1. A triple (u, w, z) of vertices is called (almost) good whenever d(u, w) = d(u, z) = 2 and µ(u, w)+µ(u, z) ≤ 2k ? 4b ₁ + 3 (and µ(u, w) + µ(u, z) = 2k ? 4b ₁ + 4). If k = 3b ₁ + γ with γ ≥ ?2, a triple (u, w, z) is almost good, and Δ = [u] ∩ [w] ∩ [z] then: either |Δ| ≤ 2; or Δ is a 3-clique and Γ is a Clebsch graph; or Δ is a 3-clique, k = 16, b ₁ = 6, and v = 31; or Δ is a 4-clique and Γ is a Schläfli graph.     

On the numerical index of real <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(µ)-spaces

We give a lower bound for the numerical index of the real space L _p (µ) showing, in particular, that it is non-zero for p ≠ 2. In other words, it is shown that for every bounded linear operator T on the real space L _p (µ), one has

$\sup \left\{ {|\int {|x{|^{p - 1}}{\rm{sign}}(x)Tx d\mu |:x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \ge {{{M_p}} \over {12{\rm{e}}}}||T||$

where \({M_p} = {\max _{t \in \left[ {0,1} \right]}}{{|{t^{p - 1}} - t|} \over {1 + {t^p}}} > 0\) for every p ≠ 2. It is also shown that for every bounded linear operator T on the real space L _p (µ), one has

$\sup \left\{ {\int {|x{|^{p - 1}}|Tx| d\mu :x \in {L_p}\left( \mu \right), ||x|| = 1} } \right\} \ge {1 \over {2{\rm{e}}}}||T||$

.

     

Commutators of Riesz transforms of magnetic Schrödinger operators

Let \({A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V}\) be a magnetic Schrödinger operator acting on \({L^2({\mathbb R}^n)}\), n ≥  1, where \({{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)}\) and \({0\leq V\in L^1_{\rm loc}({\mathbb R}^n)}\). In this paper, we show that when a function \({b\in {\rm BMO}({\mathbb R}^n)}\), the commutators [b, T _k ]f = T _k (b f) ? b T _k f, k = 1, . . . , n, are bounded on \({L^p({\mathbb R}^n)}\) for all 1 < p < 2, where the operators T _k are Riesz transforms (?/?x _k  ? i a _k )A ^?1/2 associated with A.     

On isomorphisms of some Köthe function <Emphasis Type="Italic">F</Emphasis>-spaces

We prove that if Köthe F-spaces X and Y on finite atomless measure spaces (Ω _X ; Σ _X , µ _X ) and (Ω _Y ; Σ _Y ; µ _Y ), respectively, with absolute continuous norms are isomorphic and have the property

$\mathop {\lim }\limits_{\mu (A) \to 0} \left\| {\mu (A)^{ - 1} 1_A } \right\| = 0$

(for µ = µ _X and µ = µ _Y , respectively) then the measure spaces (Ω _X ; Σ _X ; µ _X ) and (Ω _Y ; Σ _Y ; µ _Y ) are isomorphic, up to some positive multiples. This theorem extends a result of A. Plichko and M. Popov concerning isomorphic classification of L _p (µ)-spaces for 0 < p < 1. We also provide a new class of F-spaces having no nonzero separable quotient space.

     

Lie n-derivatio ns o n $ {\mathcal {}}$-subspace lattice algebras

Let \(\mathcal {L}\) be a \(\mathcal {J}\)-subspace lattice on a Banach space X over the real or complex field \(\mathbb {F}\) with dimX ≥ 3 and let n ≥ 2 be an integer. Suppose that dimK ≠ 2 for every \(K\in \mathcal {J}{(\mathcal L)}\) and \(L: \text {Alg}\, \mathcal {L}\rightarrow \text {Alg}\,\mathcal {L}\) is a linear map. It is shown that L satisfies \({\sum }_{i=1}^{n}p_{n} (A_{1}, \ldots , A_{i-1}, L(A_{i}), A_{i+1}, \ldots , A_{n})=0\) whenever p _n (A ₁,A ₂,…,A _n ) = 0 for \(A_{1},A_{2},\ldots ,A_{n}\in \text {Alg}\,\mathcal {L}\) if and only if for each \(K\in \mathcal {J}(\mathcal {L})\), there exists a bounded linear operator \(T_{K}\in \mathcal {B}(K)\), a scalar λ _K and a linear functional \(h_{K}: \text {Alg}\,\mathcal {L}\rightarrow \mathbb {F}\) such that L(A)x = (T _K A ? A T _K + λ _K A + h _K (A)I)x for all x ∈ K and all \(A\in \text {Alg}\,\mathcal {L}\). Based on this result, a complete characterization of linear n-Lie derivations on \(\text {Alg}\,\mathcal {L}\) is obtained.     

On the derivative of the Minkowski question mark function ?(<Emphasis Type="Italic">x</Emphasis>)

Let x?=?[0; a ₁ , a ₂ , …] be the regular continued fraction expansion of an irrational number x ∈ [0, 1]. For the derivative of the Minkowski function ?(x) we prove that ?′(x)?=?+∞, provided that \( \mathop {{\lim \sup }}\limits_{t \to \infty } \frac{{{a_1} + \cdots + {a_t}}}{t} < {\kappa_1} = \frac{{2\log {\lambda_1}}}{{\log 2}} = {1.388^{+} } \), and ?′(x)?=?0, provided that \( \mathop {{\lim \inf }}\limits_{t \to \infty } \frac{{{a_1} + \cdots + {a_t}}}{t} > {\kappa_2} = \frac{{4{L_5} - 5{L_4}}}{{{L_5} - {L_4}}} = {4.401^{+} } \), where \( {L_j} = \log \left( {\frac{{j + \sqrt {{{j^2} + 4}} }}{2}} \right) - j \cdot \frac{{\log 2}}{2} \). Constants κ₁, κ₂ are the best possible. It is also shown that ?′(x)?=?+∞ for all x with partial quotients bounded by 4.     

Quasi-Lipschitz equivalence of fractals

The paper proves that if E and F are dust-like C ¹ self-conformal sets with \(0 < \mathcal{H}^{\dim _H E} (E),\mathcal{H}^{\dim _H F} (F) < \infty \), then there exists a bijection f: E å F such that

$\frac{{(dim_H F)log|f(x) - f(y)|}}{{(dim_H E)log|x - y|}} \to 1$

uniformly as |x?y} å 0. It is also proved that a self-similar arc is Hölder equivalent to [0, 1] if and only if it is a quasi-arc.

     

On the Optimality of Gluing over Scales

We show that for every α>0, there exist n-point metric spaces (X,d) where every “scale” admits a Euclidean embedding with distortion at most α, but the whole space requires distortion at least \(\varOmega (\sqrt {\alpha\log n})\). This shows that the scale-gluing lemma (Lee in SODA’05: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 92–101, Society for Industrial and Applied Mathematics, Philadelphia, 2005) is tight and disproves a conjecture stated there. This matching upper bound was known to be tight at both endpoints, i.e., when α=Θ(1) and α=Θ(log?n), but nowhere in between.More specifically, we exhibit n-point spaces with doubling constant λ requiring Euclidean distortion \(\varOmega (\sqrt{\log\lambda\log n})\), which also shows that the technique of “measured descent” (Krauthgamer et al. in Geom. Funct. Anal. 15(4):839–858, 2005) is optimal. We extend this to L _p spaces with p>1, where one requires distortion at least Ω((log?n)^1/q (log?λ)^1?1/q ) when q=max?{p,2}, a result which is tight for every p>1.